Stochastic Processes and Calculus

Stochastic processes describe financial variables whose future paths are uncertain. Stochastic calculus supplies the rules used to transform and integrate those processes in continuous-time models. This branch focuses on the mathematical foundations needed to read models such as Black-Scholes and Vasicek without treating their assumptions as observed facts.

Start with the Wiener Process for the model of continuous random shocks. Continue to Ito Calculus for stochastic integration and the chain rule used with diffusion processes.

How the Concepts Connect

These tools describe a model, not a trading forecast. A diffusion may be mathematically coherent while fitting actual returns poorly.

Evidence and Assumptions to Check

• State the time unit, process, drift, volatility, and initial condition.
• Identify whether parameters use a real-world or risk-neutral probability measure.
• Check whether volatility is constant, deterministic, or itself stochastic.
• Specify correlations when several random drivers are used.
• Test sensitivity to time-step choice, calibration window, and numerical method.
• Compare model residuals with jumps, skewness, heavy tails, and volatility clustering in observed data.

Where the Models Are Used

The Black-Scholes Option Pricing Model applies stochastic calculus to an assumed stock-price diffusion. The Vasicek Interest Rate Model uses a mean-reverting diffusion for a short rate. Monte Carlo Simulation discretizes processes to approximate distributions and values numerically.

Common Mistakes

• Reading drift as a reliably forecastable return.
• Treating a Wiener increment as a percentage price change.
• Applying the ordinary chain rule and omitting Ito’s second-derivative term.
• Mixing annual volatility with daily or monthly time steps.
• Assuming continuous paths rule out material jump risk in actual markets.
• Confusing a risk-neutral pricing process with a claim about real-world probabilities.

These pages are educational introductions to model mechanics. They do not validate a particular model, price, hedge, or investment decision.

In this section

Choose a subsection first. Deeper term pages live inside each subsection, which keeps large topic hubs readable.